From dynamics on surfaces to rational points on curvespeople.math.harvard.edu/~ctm/home/text/papers/fermat/fermat.pdf · From dynamics on surfaces to rational points on curves ...

From dynamics on surfaces to rational

points on curves

Curtis T. McMullen∗

22 January, 1999

M. Jourdain: You mean when I say, ‘Nicole, bring me my slip-pers...’, I’m speaking in prose?

—Moliere, 1670.

1 Introduction

Fermat’s last theorem states that for n ≥ 3 the equation

Xn + Y n = Zn (1.1)

has no integer solutions with X, Y, Z ≥ 1. Inspiring generations of work innumber theory, its proof was finally achieved by Wiles. A qualitative result,Finite Fermat, was obtained earlier by Faltings; it says the Fermat equation hasonly a finite number of solutions (for each given n, up to rescaling).

This paper is an appreciation of some of the topological intuitions behindnumber theory. It aims to trace a logical path from the classification of sur-face diffeomorphisms to the proof of Finite Fermat. The route we take is thefollowing.

§2. The isotopy classes of surface diffeomorphisms f : S → S form the map-ping class group Mod(S). Thurston showed the elements of Mod(S) canbe classified into 3 types, depending on their dynamics: finite order, re-ducible and pseudo-Anosov. We begin by explaining a complex-analyticapproach to this classification, using the geometry of the moduli spaceMg of Riemann surfaces of genus g.

§3. An analytic family of Riemann surfaces C/B determines a classifying mapB → Mg and a monodromy map π1(B) → π1(Mg) = Mod(S). Contin-uing the study of moduli space, we sketch the Imayoshi-Shiga proof thatthere are only finitely many families (truly varying and of fixed genus)over a fixed 1-dimensional base B. A key step is to show that a family isdetermined by its monodromy.

∗Research partially supported by the NSF. 1991 Mathematics Subject Classification:11G30 (32G15, 57Mxx).

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§4. Next we present Parshin’s trick, using branched coverings, to deduce thefiniteness of sections s : B → C from the finiteness of families C/B.

§5. To begin the transition to algebra, we connect Galois theory to the fun-damental group, homology and monodromy.

§6. Finally we sketch Falting’s proof of Finite Fermat. Let C be the Riemannsurface defined by the Fermat equation (1.1). Arithmetically, we think ofthis curve as a family spread out over a base B = Spec Z − S consistingof (most of) the prime numbers. An integral solution can be reducedmod p, so it determines a section of C/B. The role of the monodromy isplayed by the action of the Galois group of Q/Q on the homology of C.By controlling the dynamics of the Galois group, we find there are justfinitely many families C/B, hence finitely many sections of C/B, hencefinitely many solutions to the Fermat equation (1.1).

In retrospect it seems Fermat, like his contemporary M. Jourdain, was unwit-tingly speaking of not just number theory but topology.

This paper is based on a lecture given at MSRI in 1997 in recognition ofThurston’s term as director, 1992-97, and inspired in part by an essay of Little-wood [Lit]. For historical accounts and detailed proofs of the developments weoutline here, see the references and notes collected at the end. I would like tothank N. Elkies, B. Mazur, R. Taylor and the referees for their help.

2 Complex analysis and dynamics on surfaces

Let S be a smooth oriented surface of genus g ≥ 0. By a simple loop α ⊂ S wemean an unoriented embedded circle that cannot be shrunk to a point. Let Sbe the set of isotopy classes of simple loops on S. The intersection pairing

i : S × S → 0, 1, 2, 3, . . .

is defined so i(α, β) is the minimum possible number of transverse intersectionsbetween representatives of α and β.

The mapping class group Mod(S) is the group of isotopy classes of orientation-preserving diffeomorphism f : S → S; the group law is composition. There is anatural action of Mod(S) on (S, i).

In this section we discuss a dynamical classification of elements of Mod(S),that sheds light on the behavior of their iterates fn : S → S.

Example: the torus. For g = 1, we have Mod(S) ∼= SL2(Z) because themapping class group acts faithfully on the homology H1(S, Z) ∼= Z2. Moreovera simple loop can be recorded by its slope in homology, giving an isomorphismbetween S and the rational points on a circle:

S = PH1(S, Q) ∼= P(Q2) ⊂ RP1 = R ∪ ∞.

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Then Mod(S) acts on S by Mobius transformations, preserving the form

i

(p

q,r

s

)=

∣∣∣∣∣det

(p r

q s

)∣∣∣∣∣ .

The automorphisms of a torus can be classified into 3 types, depending ontheir dynamical behavior.

1. Finite order. The mapping f∗ : H1(S, R) → H1(S, R) has complex eigen-values, and fn = id for some n > 0.

2. Reducible. The map f∗ has a multiple eigenvalue of ±1. Then f preservesa rational slope p/q ∈ PH1(S, R), and so it stabilizes a simple loop α ∈ S.Thus f or f2 is a Dehn twist about α.

3. Anosov. The mapping f∗ has real eigenvalues K±1, preserving a pair ofirrational slopes λ± ∈ PH1(S, R).

Using the identification S ∼= R2/Z2, the linear action of f∗ ∈ SL2(Z) givesa linear, area-preserving map F : S → S isotopic to f . In the Anosovcase, F preserves the pair of foliations F± of S by lines of slope λ±. Theleaves of F+ are stretched by a factor of K, and those of F− are shrunkby 1/K. For any α, β ∈ S, the loop Fn(α) is nearly parallel to F+, withlength comparable to Kn; thus the intersection number satisfies

i(Fn(α), β) ≍ Kn →∞ (2.1)

as n→∞.

Higher genus. Now suppose S has genus g ≥ 2. To generalize the classificationabove, let us say f ∈Mod(S) is:

2′. Reducible if there exists a finite set α1, . . . , αn ∈ S, permuted by f , withi(αi, αj) = 0; and

3′. Pseudo-Anosov if there is an expansion factor K > 1 such that the inter-section number i(fn(α), β) grows like Kn for all α, β ∈ S.

Our main goal in this section is to sketch the proof of Thurston’s result:

Theorem 2.1 (Classification of Surface Diffeomorphisms) Any mappingclass f ∈Mod(S) is either reducible, pseudo-Anosov or of finite order.

In the pseudo-Anosov case, the proof also furnishes foliations of S with isolatedsingularities, whose leaves are stretched by K±1 under a suitable representativeof f .

Remark on surface bundles. An S-bundle C → B determines a monodromyrepresentation

π1(B)→ Mod(S)

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recording the twisting of the fibers under transport around loops in the base.The simplest case is that of a 3-manifold fibering over the circle, M3 → S1;then the image of π1(S

1) is a cyclic group 〈f〉 ⊂ Mod(S). Thurston has shownM3 admits a metric of constant negative curvature iff f is pseudo-Anosov.

Riemann surfaces. The proof of the classification we present is motivated by:

Theorem 2.2 Let X be a compact Riemann surface of genus g ≥ 2. Thenthe conformal automorphism group Aut(X) is finite, and any isotopy class f ∈Mod(X) is represented by at most one conformal map.

To see this finiteness, first recall a Riemann surface of genus 2 or more ishyperbolic; that is, the universal cover X of X is isomorphic to the unit disk∆ ⊂ C. Thus X is the quotient of ∆ by a conformal action of π1(X) ⊂ Aut(∆)(Figure 1). Since Aut(∆) preserves the metric

ρ =2 |dz|

1− |z|2 (2.2)

of constant curvature −1 on ∆, we obtain a hyperbolic metric on X canonicallydetermined by its conformal structure.

The classical Schwarz Lemma states that a holomorphic map f : X → Ybetween hyperbolic Riemann surfaces can only shrink the hyperbolic metric;that is,

d(f(x), f(x′)) ≤ d(x, x′)

for all x, x′ ∈ X . In particular, Aut(X) acts by isometries.

−→

Figure 1. A surface of genus 2 covered by the hyperbolic disk.

Proof of Theorem 2.2. Let f, g : X → X be a pair of isotopic conformalmaps. Since X is compact, in the course of the isotopy points move only abounded distance D. Lifting to the universal cover, we obtain a pair of analyticmaps f , g : ∆→ ∆ with d(f(x), g(x)) ≤ D in the hyperbolic metric (2.2). Sincethe ratio of Euclidean to hyperbolic distance tends to zero at the boundaryof the disk, we have f = g on S1. But an analytic map is determined by itsboundary values, so f = g and thus f = g.

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Thus the natural map Aut(X)→ Mod(X) is injective; in particular, Aut(X)is discrete. On the other hand, Aut(X) is also compact, because it preservesthe hyperbolic metric on X ; thus the automorphism group of X is finite.

Teichmuller space. To relate Mod(S) to Aut(X), we now introduce theTeichmuller space Teich(S) of all complex structures on S.

A point in Teich(S) is specified by a Riemann surface X together with adiffeomorphism h : S → X . The map h provides a marking of X by S; forexample, it gives an identification between π1(S) and π1(X). Two markedsurfaces (h : S → X) and (g : S → Y ) define the same point in Teich(S) ifg h−1 : X → Y is isotopic to a conformal map.

The mapping-class group Mod(S) acts on Teich(S) by changing the marking;that is, f ∈Mod(S) acts by

f · (h : S → X) = (h f−1 : S → X).

The moduli space of Riemann surfaces of genus g = g(S) is the quotient space:

M(S) = Teich(S)/ Mod(S).

From the definitions it is immediate that f ·X = X in Teich(S) if and onlyif f is represented by a conformal automorphism of X . But Aut(X) is finite,so we can then conclude that f has finite order. Thus to classifying elementsf ∈ Mod(S), one is led to consider the dynamics of f on Teichmuller space.

Length functions. Next we introduce the geodesic length functions

ℓ : S × Teich(S)→ R+,

assigning to each simple loop α ⊂ S the length ℓα(X) of the geodesic repre-sentative of h(α) in the hyperbolic metric on X . These ℓα provide coordinatesmaking Teich(S) into a smooth manifold, diffeomorphic to a ball of dimension6g − 6.

By Gauss-Bonnet, the hyperbolic area of X is a constant, π(4g − 4). Thusthe only way the shape of X can degenerate is by a thin neck pinching off. Moreprecisely, letting

L(X) = infℓα(X)denote the length of the shortest geodesic loop on X , we have:

Theorem 2.3 (Mumford) For any r > 0, X : L(X) ≥ r is a compactsubset of the moduli space M(S).

A short geodesic is the core of a long, thin tube, and we have:

Proposition 2.4 There is an ǫ0 > 0 such that if:

α1, . . . , αn = α : ℓα(X) < ǫ0,

then i(αi, αj) = 0 and n ≤ 3g − 3.

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The count 3g − 3 is purely topological: it is the maximum number of disjointsimple loops on S with no pair isotopic.

Distance between Riemann surfaces. As a final ingredient to the study ofsurface diffeomorphisms, we introduce a metric on Teich(S) such that Mod(S)acts by isometries.

The Teichmuller metric is defined by

d(X, Y ) =1

2log inf

K ≥ 1 :

there exists a K-quasiconformal map

f : X → Y respecting markings

·

Here a diffeomorphism f : X → Y is K-quasiconformal if f ′ : TX → TY sendsinfinitesimal circles to ellipses with major and minor axes in ratio 1 ≤ M/m ≤K. Just as conformal maps are hyperbolic isometries, quasiconformal mapsdistort lengths of closed geodesics by a bounded factor; we have:

1

Kℓα(X) ≤ ℓα(Y ) ≤ Kℓα(X) (2.3)

when there is a K-quasiconformal map f : X → Y .

Proof of Theorem 2.1 (Classification of Surface Diffeomorphisms).Since Teich(S) is simply-connected, we can identify Mod(S) with π1(M(S)), theorbifold fundamental group of moduli space. A mapping-class f then determinesa free homotopy class of loop γ : S1 → M(S). The strategy of the proof is toseek the shortest representative of γ.

To this end, define the translation length of f by

τ(f) = infX∈Teich(S)

d(X, f ·X). (2.4)

We distinguish 3 cases.

I. τ(f) = 0, achieved. If f has a fixed-point X ∈ Teich(S), then by defi-nition the marking h : S → X transfers f to the isotopy class of a conformalautomorphism F : X → X . Since Aut(X) is finite, F has finite order, so f hasfinite order as well.

II. τ(f) > 0, achieved. Next suppose we can find a Riemann surface suchthat d(X, f ·X) = τ(f) > 0. Then the isotopy class of f can be represented by aK2-quasiconformal map F : X → X , where K = exp(τ(f)) > 1. By a theoremof Teichmuller, outside a finite set of singularities there exist complex charts inthe domain and range such that this optimal map F is an affine stretch:

F (x + iy) =x

K+ iKy.

Because d(X, f ·X) is minimized, the lines of stretch are preserved by F , defininga pair of invariant singular foliations F± whose leaves are stretched by K±1.Thus for α, β ∈ S, the loop Fn(α) is stretched along F+ as n → ∞, so theintersection number satisfies

i(Fn(α), β) ≍ Kn

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and f is pseudo-Anosov.

III. τ(f) is not achieved. Finally suppose the infimum in (2.4) is notachieved. Consider a sequence Xn ∈ Teich(S) with d(Xn, f · Xn) → τ(f).The corresponding loops γn : S1 → M(S) must exit every compact subset ofmoduli space, else we could extract a convergent subsequence. Thus by Mum-ford’s theorem the length of the shortest hyperbolic geodesic on Xn must shrinkto zero. We will show the short geodesics on Xn reveal the reducibility of themapping-class f .

Since d(Xn, f ·Xn) is bounded, there is a uniform K such that f is representedby a K-quasiconformal map

Fn : Xn → Xn.

Let ǫ = ǫ0/K3g−3, and choose n large enough such that the shortest loop γ ∈ Son Xn has length ℓγ(Xn) < ǫ.

LetA = α : ℓα(Xn) < ǫ0 ⊂ S

be the set of all isotopy classes of short loops on Xn; then |A| ≤ 3g − 3 byProposition 2.4. On the other hand, the bound (2.3) on the length distortion ofK-quasiconformal maps implies the loops

〈γ, Fn(γ), F 2n(γ) . . . , F 3g−3

n (γ)〉

have lengths less than

〈ǫ, Kǫ, K2ǫ, . . . , K3g−3ǫ = ǫ0〉,

so they all belong to A. Thus γ = F in(γ) for some 1 ≤ i ≤ 3g − 3, and hence f

is reducible.

Figure 2. The moduli space of a torus,M1 = H/SL2(Z).

The torus, reprise. For genus g = 1, Teich(S) is isomorphic to the upperhalf-plane H with Mod(S) ∼= SL2(Z) acting by Mobius transformations. Themoduli spaceM1 = H/SL2(Z) is the (2, 3,∞) orbifold, a sphere with one cuspand two cone-points (Figure 2). Every nontrivial loop onM1 is either

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• represented by a geodesic,

• homotopic to a cone point, or

• homotopic to a short loop around the cusp.

These possibilities correspond to the Anosov, finite order, and reducible mapping-classes respectively.

3 Families of Riemann surfaces

In this section we will show an argument similar to the classification of surfacediffeomorphisms leads to a proof of:

Theorem 3.1 (Geometric Shafarevich conjecture) For a given base B andgenus g ≥ 2, there are only finitely many truly varying families C/B with fibersof genus g.

Definitions. By a family C/B we mean the data of:

• a 1-dimensional complex manifold B, connected, forming the base of thefamily; and

• a 2-dimensional complex manifold C, the total space, equipped with aholomorphic fibration π : C → B; such that

• the fibers Ct = π−1(t) are compact Riemann surfaces of genus g; and

• B is isomorphic to B − P , the complement of a finite set P (possiblyempty) in a compact Riemann surface B.

A family is locally constant if Ct∼= Cu for all t, u ∈ B; otherwise it is truly vary-

ing. We regard two families C, C′ over B as the same if there is an isomorphismC ∼= C′ respecting the projections to B.

The proof of finiteness we will sketch, due to Imayoshi and Shiga, is basedon a Schwarz Lemma for moduli space.

Complex geometry of Mg. Let Mg denote the moduli space of Riemannsurface of genus g ≥ 2; Tg its universal covering, Teichmuller space; and Modg =π1(Mg) the mapping-class group.

There is a natural complex structure on Mg such that any family C/Bdetermines a holomorphic classifying map

F : B →Mg,

defined by F (t) = [Ct], and a monodromy representation

F∗ : π1(B, t)→ Mod(Ct) ∼= π1(Mg),

recording the twisting of the fiber under transport around closed paths in thebase.

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With the complex structure lifted from Mg, Tg is isomorphic to an open,

bounded domain in C3g−3. Lifting F to the universal cover B of B, we obtaina bounded analytic map

F : B → Tg ⊂ C3g−3.

Thus if B is isomorphic to C or C, then F must be constant and therefore allfamilies C/B are trivial.

Now consider the case where B is hyperbolic, i.e. B ∼= ∆. Even in this case,the boundedness of Tg controls the geometry of maps F : ∆→ Tg. This controlis made precise by Royden’s theorem:

Theorem 3.2 (Modular Schwarz Lemma) Any holomorphic map F : B →Mg is distance-decreasing from the hyperbolic metric on B to the Teichmullermetric on Mg. In fact d(F (s), F (t)) ≤ d(s, t)/2.

We regard this theorem as a Schwarz Lemma for maps with targetMg.

Example: Monodromy over ∆∗. Let C/∆∗ be a family of Riemann surfacesof genus g over the punctured disk ∆∗ = t : 0 < |t| < 1. In the hyperbolicmetric, ∆∗ has a cusp at t = 0; as r tends to zero, the hyperbolic length ofthe circle S1(r) also tends to zero. The classifying map F : ∆∗ →Mg shrinksdistances, so the generator of the monodromy group

F∗(π1(∆∗)) = 〈f〉 ⊂ Modg

has translation length τ(f) = 0. By the classification of surface diffeomorphisms,f is reducible or of finite order. Thus a finite iterate fn is simply a product ofDehn twists (a classical observation of Lefschetz).

Proof of Theorem 3.1 (Geometric Shafarevich Conjecture). Mimickingthe proof of finiteness of Aut(X), we will show there are only finitely many trulyvarying families C/B by showing the space F of all classifying maps F : B →Mg is compact and discrete.

I. Irreducibility. Fix a basepoint t ∈ B. As for a mapping-class, we say C/Bis reducible if the monodromy group

H = F∗(π1(B, t)) ⊂Mod(Ct)

preserves a collection of disjoint simple loops α1, . . . , αm on Ct.Suppose C/B is reducible, and let α be simple loop on Ct with finite mon-

odromy. After passing to a finite covering of B, we can assume α is invariantunder π1(B). By a theorem of Wolpert, the geodesic length L(u) = ℓα(Cu)(now globally well-defined) is subharmonic; that is ∆u ≥ 0. By the maximumprinciple, L(u) is constant on B.

Consider the smallest convex subsurface D ⊂ Ct carrying all loops suchas α with finite monodromy. Using the rigidity of D and an argument withquasifuchsian groups, one can show that ∂D = ∅. Thus D = Ct, and the familyC/B is trivial.

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Summing up, a truly varying family C/B is irreducible.

II. Compactness. To show F is compact, we begin by showing the basepointF (t) must lie in a compact subset of moduli space. By Mumford’s theorem, itsuffices to find ǫ > 0 such that the length of the shortest geodesic on [Ct] = F (t)satisfies L(Ct) ≥ ǫ.

Choose a finite set of closed paths γ1, . . . , γn on B generating π1(B, t).By Royden’s theorem, there is a uniform bound K ≥ 1 (depending only onmax ℓ(γi)) such that the monodromy of C/B around γi is represented by aK-quasiconformal map fi : Ct → Ct.

Let ǫ = ǫ0/K3g−3. Suppose the shortest loop α1 on Ct has length less thanǫ, and let α1, . . . , αm enumerate all loops on Ct shorter than ǫ0, in order ofincreasing length. Since m ≤ 3g − 3 there is an index p such that

Kℓαp(Ct) < ℓαp+1

(Ct).

Now fi changes the lengths of loops by at most a factor of K (by (2.3), so fi mustpermute the loops α1, . . . , αp. Since 〈fi〉 generates the full monodromy groupF∗(π1(B)), the system of curves α1, . . . , αp is invariant over the entire baseB, and thus C/B is reducible. Hence C/B is trivial, contrary to our assumptionthat C/B is a truly varying family.

Thus F (t) lies in the compact set X ∈ Mg : L(X) ≥ ǫ for all F ∈ F . SinceF : B →Mg is distance decreasing, the family F is bounded and equicontinuouson each compact subset of B. Thus any sequence has a convergent subsequence,and hence F is compact.

III. Discreteness. Finally we show F is discrete. If F, G ∈ F are close enough,then they are homotopic. We will show this implies F = G.

Since F and G are homotopic, there are lifts

F , G : ∆→ Tg ⊂ C3g−3

withsupt∈∆

d(F (t), G(t)) ≤ D (3.1)

in the Teichmuller metric on Tg.

By a theorem of Fatou, a bounded analytic function such as F has well-defined boundary values F (t) for almost every t ∈ S1. In a truly varying family,

the boundary values lie in ∂Tg, since the image F (∆) is properly embedded inTeichmuller space.

Now ∂Tg contains a countable union of complex hypersurfaces A, parame-terizing curves with nodes that arise as limits of curves of genus g. Along agiven component of A, the nodes are marked by a set of simple closed curves onS. But the monodromy F∗(π1(B)) is irreducible, so there are no distinguished

simple closed curves on S, and thus (almost all) the boundary values of F lie in∂Tg −A.

On the other hand, one knows that the ratio of the Euclidean metric on Cn

to the Teichmuller metric on Tg tends to infinity at points in ∂Tg−A, so by (3.1)

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we have F (t) = G(t) for almost every t ∈ S1. Since a holomorphic function isdetermined by its boundary values, we have F = G.

IV. Finiteness. Having shown F is compact and discrete, we conclude thatthe set of potential classifying maps for C/B is finite. The classifying mapF : B → Mg determines the family C/B up to finitely many choices (limitedby |Hom(π1(B), Aut(Ct))|), and hence there are only finitely many truly varyingfamilies C/B of genus g.

From the proof we also record:

Corollary 3.3 (Rigidity) A truly varying family C/B is determined up tofinitely many choices by its monodromy F∗ : π1(B)→ Modg.

4 Branched covers

In this section we present Parshin’s argument bounding the number of holomor-phic sections of a family C/B in terms of the number of families D/B of highergenus. (The same construction was used by Kodaira to construct truly varyingfamilies over a compact base.)

Theorem 4.1 Given a genus g ≥ 1 and a base B, there exists a genus h ≥ 2and a finite-to-one map

Families C/B with fibers of genus g,

equipped with sections s : B → C

→

Families D/B

with fibers of genus h

.

For each t ∈ B, the surface Dt is a covering of Ct branched over the single points(t) ∈ C(t).

Here we regard (C, s) and (C′, s′) as the same if there is an isomorphismC ∼= C′ over B sending s to s′.

Proof. Given a pointed topological surface (S, p) of genus g, we can form thecovering space

π : T → (S, p)

corresponding to the kernel of the map

π1(S, p)→ H1(S, Z/2) ∼= (Z/2)2g.

Letting P = π−1(p), we can similar form the branched covering U → T corre-sponding to the map

π1(T − P )→ H1(T − P, Z/2).

Since |P | > 1, any small loop around a point of P is nonzero in H1(T −P, Z/2),and hence U → T is branched over every point in P .

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The composite branched covering U → (S, p) is canonical, so the corre-sponding family of branched coverings Dt → (Ct, s(t)) fit together to form afamily D/B. The original family C/B is a quotient of D/B by a subgroup ofAut(D/B), and the graph of the section, s(B) ⊂ C, corresponds to the fixed-point set of an element of Aut(D/B). Since Aut(D/B) is finite, the family D/Bdetermines C/B and s : B → C up to finitely many choices.

−→

Figure 3. A surface of genus 3 is a double cover of a torus branched along 4 points.

Example. Let (S, p) = (R2/Z2, 0) be a flat torus. Then (T, P )→ (S, p) coversS by degree 4, with P the points of order 2 on T . The covering U → T , branchedover P , has genus 33 and deck group (Z/2)5; it is the compositum of 5 doublecoverings, each of the form shown in Figure 3.

Corollary 4.2 (Geometric Mordell Conjecture) A truly varying family C/Bof genus g ≥ 2 has only a finite number of sections s : B → C.

Proof. For each section s : B → C, we can form the family D/B of genus h ≥ 2branched over s(B). The number of truly varying families D/B is finite, so thenumber of pairs (C, s) is finite up to automorphism over B. But Aut(C/B) isfinite, so the number of sections s : B → C is finite as well.

5 Monodromy over a field

To begin the passage from geometry to arithmetic, in this section we discussrelatives of the fundamental group, homology and monodromy that can be con-structed via algebraic geometry.

Valuations. Let B be a compact Riemann surface. Algebraically, B is specifiedby its field of meromorphic functions K = K(B).

A (discrete) valuation on K is a surjective homomorphism

v : K∗ → Z,

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defined on the multiplicative group of K and satisfying v(f+g) ≥ min(v(f), v(g)).All valuations are of the form

vp(f) = ordp(f),

where ordp(f) = n (or −n) if f has a zero (or pole) of order n at p. Thus thepoints of B can be recovered as the valuations of K.

Algebraic π1. Let K denote the algebraic closure of K. The Galois groupGal(K/K) is a first approximation to an algebraic version of π1(B).

To describe this Galois group topologically, recall that the profinite comple-tion of a group G is the inverse limit

G = lim←−

G/N

over all normal subgroups N of finite index. For any space E, let π1(E) denotethe profinite completion of π1(E).

Every algebraic extension of K is of the form K ′ = K(B′), where B

′ → B

is a finite covering branched over a finite set P ⊂ B. Since B′is determined by

a subgroup of finite index in π1(B − P ), we can take the limit over all possibleP and obtain:

Gal(K/K) = lim←−

P

π1(B − P ).

Thus the algebraic closure K detects the fundamental group of B ‘puncturedeverywhere’.

Ramification. To construct a Galois group closer to π1(B), let us say a fieldextension K ′/K of degree d is ramified over a valuation vp if there are fewerthan d extensions of vp to a valuation v′ on K ′. The ramified valuations vp

correspond exactly to the branch points p of B′ → B.

Let KP ⊂ K denote the field generated by all finite extensions of K ramifiedonly over vp, p ∈ P ⊂ B. Then for any finite set P ⊂ B we have

Gal(KP /K) ∼= π1(B − P ). (5.1)

In particular we can recover π1(B) as a Galois group by taking P = ∅.Monodromy. Let C/B be a family over a base B = B − P . With the Galois-theory version (5.1) of π1(B) in hand, we now turn to the construction of analgebraic relative of the monodromy map

F∗ : π1(B, t)→ Mod(Ct).

The first step is to retreat from the mapping-class group, by replacing π1(Ct)with the family of groups H1(Ct, Z) ∼= Z2g. We then obtain a linear represen-tation

ρ : π1(B, t)→ Aut(H1(Ct, Z)) = GL2g(Z), (5.2)

recording the twisting of homology around loops on the base.

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Next fix a prime ℓ, and let Zℓ = lim←−

Z/ℓn denote the ℓ-adic integers. Note

that any prime p 6= ℓ becomes invertible in ℓ; the local ring Zℓ is the completionof Z at ℓ. By reducing mod ℓn we obtain a system of finite representations

ρℓn : π1(B)→ GL2g(Z/ℓn)

that fit together to determine a map

ρℓ : Gal(KP /K) ∼= π1(B) → GL2g(Zℓ).

This ℓ-adic Galois representation is nothing more than the completion at ℓof the monodromy on integral homology (5.2). Our goal is to reconstruct ρℓ

using the methods of algebraic geometry.

The Jacobian. The first step is to use the Jacobian to build an algebraicversion of the homology of C.

Let X be a compact Riemann surface of genus g ≥ 1. A divisor D =∑ap · p ∈ Z[X ] is a finite formal sum of points of X ; its degree is

∑ap. A

principal divisor is one of the form

D = (f) =∑

p

vp(f) · p,

where f ∈ K∗(X) is a meromorphic function and vp(f) is the valuation of f atp. The Jacobian of X is the quotient

Jac(X) = Div0(X)/(f) : f ∈ K∗(X)

of divisors of degree zero by principal divisors.By a theorem of Abel, the Jacobian is a complex projective torus, or Abelian

variety, alternatively described as:

Jac(X) = Ω(X)∗/H1(X, Z) ∼= Cg/Λ.

Here Ω(X) is the vector space of holomorphic 1-forms θ on X ; these pair linearlywith cycles γ ∈ H1(X, Z) by

〈γ, θ〉 =

∫

γ

θ ∈ C.

Fixing q ∈ X , there is a natural embedding

X → Jac(X)

sending p to the divisor p− q, and inducing an isomorphism on homology:

H1(X, Z) ∼= H1(Jac(X), Z). (5.3)

Moreover the map Xg → Jac(X) given by (pi) 7→∑

(pi− q) is surjective. Usingthis surjectivity one can derive algebraic equations for Jac(X) as a projectivevariety in terms of equations for X .

14

For any Abelian variety A of dimension g, let A[n] ∼= (Z/n)2g denote itspoints of order n. Then from (5.3) we have

H1(X, Z/ℓn) ∼= Jac(X)[ℓn].

Since the group law on Jac(X) is algebraic, the finite set Jac(X)[ℓn] is definedby a system of algebraic equations and can be constructed without reference tothe topology of X . This is the desired algebraic version of homology.

Families of Abelian varieties. We can now associate to each family C/B afamily of Abelian varieties A/B with At = Jac(Ct). For any n ≥ 0, the familyof groups At[ℓ

n] ∼= (Z/ℓn)2g becomes trivial after passing to a finite covering ofthe base, and thus the deck transformations determine a representation

ρℓn : π1(B)→ Aut A[ℓn] = GL2g(Z/ℓn).

Since the target is finite, we obtain a system of representations of π1(B) ∼=Gal(KP /K), K = K(B), compatible under the map

A[ℓn+1]→ A[ℓn]

given by multiplication by ℓ. Passing to the limit as n→∞, we obtain a purelyalgebraic construction of the ℓ-adic representation

ρℓ : Gal(KP /K)→ GL2g(Zℓ). (5.4)

This is the desired algebraic version of monodromy.

Moduli of Abelian varieties. For one final perspective on (5.4), we recallthat the moduli space of (principally polarized) Abelian varieties is given bythe quotient Ag = Hg/Sp2g(Z), where Hg is the Siegel upper halfspace of g × gsymmetric complex matrices Z with Im(Z) positive definite.

From a family C/B we obtain maps

BF→Mg

Jac→ Ag

whose composition Jac F classifies the family A = Jac(C)/B. Passing to profi-nite fundamental groups, we obtain a sequence of successively coarser mon-odromy representations:

Gal(KP /K) ∼= π1(B) → π1(Mg) → π1(Ag) → GL2g(Zℓ).

The first map records the action of the Galois group of the base on all finitecovers of the fibers; the last, on the ℓ-adic homology of the fibers. This lastrepresentation is the ℓ-adic monodromy ρℓ.

6 Finite Fermat

In this section we finally sketch the proof of:

15

Theorem 6.1 (Finite Fermat) For n ≥ 4, the equation

Xn + Y n = Zn (6.1)

has only a finite number of integral solutions with gcd(X, Y, Z) = 1.

In brief, the idea of the proof is to think of the Riemann surface C ⊂ P2(C)defined by (6.1) as a family C/B spread out over the prime numbers p ∈ Z. Thefiber Cp is the reduction of C mod p. An integral solution to the Fermat equationgives a coherent family of points on each fiber Cp, and hence a section of C/B.The condition n ≥ 4 implies the genus bound g(C) ≥ 2; thus Finite Fermatfollows from an arithmetic version of the finiteness of sections for families C/B.

To understand the sense in which the Fermat equation determines a familyC/B, we begin with a study of the base.

Spec Z. The spectrum of the ring of integers Z is the space

Spec Z = 0, 2, 3, 5, 7, 11, . . .

consisting of the prime numbers p ∈ Z, plus the ‘generic point’ 0. This spacecomes equipped with a topology and a sheaf O that plays the role of the sheafof holomorphic functions on a Riemann surface. The global sections of O are Z

itself, and the field of meromorphic functions becomes K(B) = Q.One can also see the prime numbers as points by considering valuations on

K(B) = Q. Indeed, every valuation v : Q∗ → Z is of the form

vp(r/s) = ordp(r) − ordp(s)

for some prime number p ∈ Z, where ordp(r) is the largest n such that pn dividesr.

The shape of a prime. A valuation v on a field K determines:

• A local ring Ov = f : v(f) ≥ 0 ⊂ K;

• The maximal ideal mv = f : v(f) > 0 in Ov;

• A residue field k = Ov/mv; and

• A local fundamental group G = Gal(k/k) .

For example, if B is a compact Riemann surface, K = K(B) and v = vp forp ∈ B, then Ov is the ring of meromorphic functions analytic at p, and the map

Ov 7→ Ov/mv = k ∼= C

is simply the point evaluation f 7→ f(p). Since C is algebraically closed, thelocal fundamental group G at p is trivial, reflecting the contractibility of p.

16

In contrast, a prime p ∈ Z behaves more like a circle than a point. For thecorresponding valuation v = vp on K = Q we have

Ov = Z(p) = r/s : (s, p) = 1mv = pZ(p)

k = Fp∼= Z/pZ

G = Gal(Fp/Fp) = Z.

The local fundamental group G = Z (the profinite completion of Z) is (topo-logically) generated by the Frobenius automorphism σp of Fp sending x to xp.

Since π1(S1) = Z as well, we are led to picture a prime as a topological circle.

The fibers of C. Let C/B be a family over a Riemann surface B. Then thefiber Cp over p ∈ B can be described as the reduction of C to a curve over theresidue field C = Op/mp.

Similarly, if C ⊂ P2 is a plane curve defined by a homogeneous equationF (X, Y, Z) ∈ Z[X, Y, Z], then for each prime p ∈ Spec Z we can reduce F modp to obtain a curve Cp in the projective plane over Fp. We say C has goodreduction at p if Cp is smooth.

Loops in the base. When C is smooth to begin with, it has good reductionoutside a finite set of primes S; removing these points from the base, we obtaina family with smooth fibers Cp, p 6∈ S. Thus the natural base for C is

B = (Spec Z)− S = Spec S−1Z.

By analogy with (5.1), the fundamental group of the base is

π1(B) = Gal(QS/Q),

where QS is the maximal algebraic extension of Q unramified outside S.For example, the Fermat curve is given in the affine chart Z 6= 0 by the

equation f(x, y) = xn + yn = 1, with differential

df(x, y) = nxn−1dx + nyn−1dy.

So long as p does not divide n, the equation f = df = 0 has no solutions in F2

p

and Cp is smooth. On the other hand, df vanishes identically when p divides n,so the prime divisors of n give exactly the points of bad reduction S.

For each prime p 6∈ S, the Frobenius σp ∈ Gal(Fp/Fp) lifts to an elementof Gal(QS/Q), well-defined up to conjugacy, that we also denote by σp. By atheorem of Cebotarev, the Frobenius elements are dense in Gal(QS/Q). Onecan picture a prime p as a loop in the base B = (Spec Z) − S, and σp as thecorresponding element in π1(B). Monodromy around these ‘prime loops’ playsa crucial role in the study of families defined over Q.

Homology and monodromy. Let C be a curve of genus g defined over Q,with smooth fibers over B = Spec Z − S. Fix a prime ℓ and replace S with

17

S∪ℓ. Then the Jacobian A = Jac(C) and the map x 7→ ℓx on A have naturaldefinitions over Q with good reduction outside S.

From the geometric theory of the Jacobian, discussed earlier, there is anatural isomorphism

A[ℓn] = H1(C, Z/ℓn).

At the same time A[ℓn] has coordinates lying in a finite extension of Q, so weget an action of the Galois group on the homology of C. Taking the limit asn→∞, we obtain the ℓ-adic Galois representation

ρℓ : π1(B) = Gal(QS/Q) → AutH1(C, Zℓ) ∼= GL2g(Zℓ).

This linear action of π1(B) on the homology of C is the arithmetic version ofthe monodromy.

Solutions to Fermat’s equation and branched covers. Now supposewe have relatively prime integers (X, Y, Z) (such as (1, 0, 1)) solving Fermat’sequation Xn + Y n = Zn. Then P = [X : Y : Z] ∈ P2(Q) gives a rational pointon the Fermat curve C. Using Parshin’s trick, we obtain a Riemann surface Dwith a covering D → C branched over P , with the genus of D controlled bythat of C. By studying this covering arithmetically, one can show D is definedover a finite extension K/Q with good reduction outside a finite set of primesS, where (K, S) depends only on C. Finally D determines the rational pointP ∈ C up to finite ambiguity.

Thus to prove Theorem 6.1(Finite Fermat), it suffices to establish:

Theorem 6.2 (Arithmetic Shafarevich conjecture) Fix a number field K,a finite set of primes S of K and a genus g ≥ 2. Then there are only finitelymany curves C of genus g defined over K with good reduction outside S.

Sketch of the proof. For concreteness we treat the case K = Q. Let A =Jac(C), adjoin ℓ to S and let

ρℓ : Gal(QS/Q)→ GL2g(Qℓ)

be the monodromy representation. Let σp ∈ Gal(QS/Q) be a lift of the Frobe-nius for each prime p 6∈ S. Since the conjugacy class of σp is determined by p,the trace

Tr(σp) = Tr(ρℓ(σp))

is well-defined.The strategy of the proof is to show that there are only finitely many possi-

bilities for ρℓ, and that each determines A and hence C up to finite ambiguity.We will sketch the main steps, and indicate their resonance with ideas in thegeometric proof of §3.

1. Semisimplicity. The representation ρℓ is semisimple. In particular, ρℓ

is determined (up to conjugacy) by its trace

Tr ρℓ : Gal(QS/Q)→ Zℓ.

18

This semisimplicity is like the irreducibility of the monodromy

F∗ : π1(B)→ Mod(S)

demonstrated in §3. For example, if F∗(π1(B)) were reducible, generated byDehn twists about disjoint, homologically nontrivial loops, then ρℓ would beunipotent rather than semisimple, with the same trace as the trivial represen-tation.

2. Finite generation. There is a finite set of primes T disjoint from Ssuch that the traces 〈Tr(σp) : p ∈ T 〉 determine ρℓ.

This statement is similar to the finite generation of π1(B). One first showsthat different representations can be distinguished over an extension K/Q ofdegree d ≤ d(ℓ, g). According to Hermite, there are only finitely many such ex-tensions unramified outside S; and by Cebotarev, there is a finite set of primes Tsuch that for each K, 〈σp : p ∈ T 〉 represents every conjugacy class in Gal(K/Q).The traces of these σp then determine ρℓ.

3. The Weil bounds. For any prime p 6∈ S, the trace of the Frobenius liesin Z and obeys a bound |Tr(σp)| ≤ N(p, g) independent of C.

Weil showed the number of points on a curve over a finite field can becomputed by applying the Lefschetz fixed-point formula to the Frobenius; moreprecisely,

|C(Fp)| =∑2

i=0(−1)i Tr(σp|Hi(C, Qℓ))

= 1− Tr(σp|H1(C, Qℓ)) + p.(6.2)

Each trace above actually lies on Z. Since

H1(C, Qℓ)∗ = H1(C, Zℓ)⊗Qℓ,

we see the trace of the monodromy, Tr(σp), is the same as the middle termin the Lefschetz formula, namely the trace of the Frobenius on H1. On theother hand, |C(Fp)| is bounded above by Riemann-Roch; there is a rationalmap C(Fp)→ P1(Fp) of degree 1 < d = O(g), so we have

|C(Fp)| ≤ d(g)|P1(Fp)| = O(gp).

By (6.2), we have |Tr(σp)| = O(gp) as well. (The Riemann hypothesis for curvesover finite fields, also proved by Weil, gives the sharper bound |Tr(σp)| ≤ 2g

√p.)

Combining (1–3), we deduce:

Only finitely many representations ρℓ occur for fixed (S, g).

Remark: Lengths of primes. The Weil bound is reminiscent of the ModularSchwarz Lemma (Theorem 3.2), i.e. the contracting property for holomorphicmaps F : B → Mg. For example, when g = 1 each α ∈ SL2(Z) determines aclosed loop onMg whose Teichmuller length satisfies

2 cosh(ℓα(Mg)) = |Tr(α)|.

19

Since F shrinks the hyperbolic metric on B by at least a factor of two, forα ∈ π1(B) we obtain the a priori bound:

|Tr(F∗α)| ≤ 2 cosh(ℓα(B)/2).

Is the Weil bound perhaps a measurement of the ‘hyperbolic length’ of theloop represented by a prime?

4. Isogeny. The representation ρℓ determines A up to isogeny over Q.A homomorphism φ : A→ B between Abelian varieties of the same dimen-

sion is an isogeny if its kernel is finite. If |Ker(φ)| is relatively prime to ℓ, thenφ induces an isomorphism

A[ℓn]→ B[ℓn]

for every n, and hence an isomorphism between the ℓ-adic representations ρℓ

for A and B. Hence the best one can hope for is that ρℓ determines A up toisogeny, and indeed this is the case.

This result is an arithmetic version of the rigidity of families C/B, i.e. thefact that a truly varying family is determined by its monodromy (Corollary 3.3).

5. Heights. Given A, there is an upper bound h(B) ≤ h0 on the height ofany Abelian variety B isogenous to A over Q.

Let Ω(A) be the 1-dimensional vector space of holomorphic sections of thecanonical bundle of A = Cg/Λ. Any θ ∈ Ω(A) lifts to a constant form

θ = C dz1 · · · dzg

on Cg. The arithmetic structure of A determines an additive subgroup of integralg-forms,

Ω(A)Z = Zθ0 ⊂ Ω(A).

The intrinsic height h(A) measures the volume of the complex manifold A(C)with respect to the minimal integral form:

h(A) = −1

2log

(1

2g

∫

A(C)

|θ0|2)

.

The pulled-back volume increases under an isogeny π : A→ B:∫

A

|π∗(θ0)|2 = deg(A/B)

∫

B

|θ0|2;

however in compensation one usually finds a shift in the minimal integral form,

[Ω(A)Z : π∗Ω(B)Z] =√

deg(A/B),

and thus h(A) = h(B). Taking into account the less usual isogenies, one stillobtains a bound h(B) ≤ h0 depending only on A.

In the case of Riemann surfaces, a natural height on the moduli spaceMg isgiven by h(X) = − logL(X), where L(X) is the length of the shortest geodesic.To replace Mumford’s theorem that

Mg(ǫ) = X : L(X) ≥ ǫ > 0

20

is compact, we have:6. Finiteness. The set of Abelian varieties A/Q with height h(A) ≤ h0 is

finite.Let us define the naive height of A/Q by

H(A) = log max|p1|, |q1|, . . . , |pn|, |qn|,

where (pi/qi)ni=1 are the rational numbers appearing in suitable equations defin-

ing A. It suffices to show H(A) is controlled by h(A).To convey the idea of the result, suppose A is the elliptic curve with equation

y2 = x(x− 1)(x− p/q),

0 < p/q < 1/2. Then H(A) = log q. Clearing denominators, we obtain theminimal integral form

θ0 =dy

x(x − 1)(qx− p)

on A, with volume

∫

A

|θ0|2 = 2

∫

C

|dx|2|x(x − 1)(qx− p)| ≍

log(q/p)

q2≤ log q

q2·

The intrinsic height is essentially log(1/ vol(A)), so we find h(A) ≍ log q ≍H(A). Thus a bound on h(A) pins A down to a finite set.

The general case entails the delicate construction of an arithmetic modulispace for higher-dimensional Abelian varieties.

Combining (4–6), we deduce:

Only finitely many Abelian varieties A/Q correspond to a givenGalois representation ρℓ.

7. Polarizations. The last step in the proof is to show: The curve C isdetermined by the Abelian variety A = Jac(C) up to finitely many choices.

The intersection pairing of topological 1-cycles on C gives a symplectic form

ω : ∧2H1(A, Z) = ∧2H1(C, Z) → Z.

The form ω is called a principal polarization of A; it determines a flat metricmaking A into a Kahler manifold of total volume 1.

The classical Torelli Theorem states that C is uniquely determined by thepair (A, ω). To show A alone determines C up to finite ambiguity, it suffices toshow the set of principal polarizations falls into finitely many orbits under theaction of Aut A.

To give an idea of the proof, we sketch instead a related result for a real torusT = Rn/Zn. Instead of principal polarizations, we consider positive definitequadratic forms q : H1(T, Z)→ Z determining flat metrics on T of total volume1. We will show such q fall into finitely many orbits under the action of Aut T .

21

The set of all pairs (T, q) determines a discrete set Q ⊂M, where

M = SOn(R)\SLn(R)/SLn(Z)

is the moduli space of flat tori of volume 1. On the other hand, because q isan integral form, the q-length of the shortest geodesic loop satisfies L(T, q) ≥ 1.By a result of Mahler (a precursor to Mumford’s theorem),

(T, g) ∈M : L(T, g) ≥ 1

is compact, so Q is finite. Thus there are only finitely many possibilities for qup to the action of AutT .

Conclusion. Combining (1-7), we deduce that only a finite number of curvesC/Q of genus g have good reduction outside S.

The methods can be generalized to an arbitrary number field K.

With the Shafarevich conjecture in hand, Parshin’s covering trick yields:

Corollary 6.3 (Arithmetic Mordell’s Conjecture) A smooth curve C ofgenus g ≥ 2 defined over a number field K has only a finite number of K-rational points.

In particular, the Fermat equation with n ≥ 4 has only a finite number ofsolutions.

More on arithmetic topology. The parallel approaches to the Shafarevichconjectures, detailed above, are but one instance of the interplay between topol-ogy, complex geometry and number theory. We conclude by mentioning a fewmore examples and references.

1. The action of the Galois group Gal(Q/Q) on the homology of a curveC can be refined to a representation into the arithmetic mapping-class

group Mod(C). This mixture of Galois theory and topology, especially inthe case C = P1 − 0, 1,∞, forms the basis for Grothendieck’s theory of‘dessins d’enfants’, and has been the subject of much recent activity [Sn].

2. The intrinsic height of an abelian variety, formulated above, is an exampleof Arakelov geometry, which aims to treat the finite and infinite places ofa field on an equal footing [La2], [So].

3. The analogy between number fields and functions fields of Riemann sur-faces goes back at least to Weil [Wl2].

4. A dictionary between complex-analytic Nevanlinna theory and Diophan-tine approximation has been developed by Vojta, leading to another proofof Mordell’s conjecture [Voj1], [Voj2].

5. Mochizuki has developed the methods of Teichmuller theory in the p-adicsetting [Mo].

22

6. Finally we mention that Mazur and others have suggested one shouldpicture the primes p ∈ Spec Z not just as loops, but as knots in a 3-sphere. Class field theory then parallels the investigation of the homologyof branched covers of S3, and Iwasawa theory provides the analogue of theAlexander polynomial for a prime.

In support of the correspondence

Spec Z ←→ S3,

note that Spec Z is ‘simply-connected’ (there are no unramified extensionof Q), and a ‘homology 3-sphere’ (Hp(Spec Z, Gm) = 0 except in dimen-sions p = 0 and p = 3 [Maz1, p. 538]).

Ideas from number theory can also inform research on 3-manifolds; forexamples, see Reznikov’s papers on ‘arithmetic topology’ [Rez1], [Rez2,§14].

7 Notes

§1. The proof of Fermat’s last theorem appears in [Wi], [TW]; for surveys, see[RS], [Ri] and [DDT].

§2. Thurston’s classification of surface diffeomorphisms is outlined in [Th1] anddeveloped in detail in [FLP]; here we present Bers’ complex-analytic approach[Bers]. Mumford’s compactness theorem appears in [Mum]; for a related resultdue to Weil, see [Wl1]. For more about the hyperbolic geometry of surfaces, seeBuser’s text [Bus].

The theme of short geodesics, appearing here in the proofs of the classifi-cation of surface diffeomorphisms and of the geometric Shafarevich conjecture,is also seen in Thurston’s work on rational maps and hyperbolic 3-manifoldsvia iteration on Teichmuller space [Mc1]. The theory of hyperbolic 3-manifoldsfibering over the circle is presented in [Th2] and [Ot]; see also [Mc2, §3], [Br].

We remark that by Mostow rigidity and the Hyperbolization Conjecture,the mapping-class group Mod(M3) is expected to be finite for most closed 3-manifolds.

§3. The geometric Shafarevich conjecture was proved by Arakelov, generalizingParshin’s treatment of the case of a compact base B [Ar], [Par]. The proofsketched here is a slight variant of that by Imayoshi and Shiga [IS]; we useWolpert’s result that ℓα(X) is subharmonic [Wol].

A systematic introduction to the complex geometry of Teichmuller space isprovided by the texts [Gd], [IT], [Le], and [Nag]. These books present Bers’model for Teichmuller space as a bounded domain, and cover Royden’s theoremthat the Teichmuller and Kobayashi metrics coincide.

The finiteness of families C/B also holds for genus g = 1 if we require thatC admits a section s : B → C. Without a section, there may be infinitelymany families of elliptic curves C/B for a given classifying map F : C →M1.

23

Similarly one can construct infinitely many curves C/Q of genus 1 with a fixedlocus of bad reduction (and C(Q) = ∅); see [Maz2, p.241].

§4. The geometric Mordell conjecture was formulated by Lang and proved byManin [La1, p. 29], [Man]; see also [Gr]. Parshin’s trick appears in [Par]. Thesame construction was used by Kodaira to exhibit truly varying families C/Bover a compact base [Ko].

For an alternate proof of the geometric Mordell conjecture, one can repeatthe argument of §3 using the fact that a section of C/B determines a classifyingmap F : C/B →Mg,1 to the moduli space of pointed Riemann surfaces of genusg.

Deligne proved quite generally that for any smooth complex projective familyof varieties V/B, the number of possible linear monodromy representations ofdimension n for a fixed B is finite [De2]. More precisely, if t ∈ B is a basepointand dimHi(Vt, Q) = n, then there are at most N(B, n) possibilities for the map

ρ : π1(B, t)→ GL(Hi(Vt, Q)) ∼= GLn(Q)

up to conjugacy.

§5. A detailed treatment of the Jacobian of a Riemann surface can be found inGriffiths and Harris [GH].

§6. The arithmetic conjectures of Mordell and Shafarevich were proved byFaltings [Fal]. Our sketch follows Deligne’s presentation [De1]. See also [Sz] (es-pecially §5.2 on the arithmetic of the covering D → C). More about arithmeticon curves, leading up to Falting’s theorem, can be found in Mazur’s survey[Maz2] and the collection [CS].

Work on the Weil conjectures is surveyed in [Ka]. The finiteness of thenumber of principal polarizations of a given abelian variety is proved in [NN];see also Milne’s article [CS, Ch. V, §18].

References

[Ar] S. Arakelov. Families of algebraic curves with fixed degeneracies. Math.USSR Izv. 35(1971), 1269–1293.

[Bers] L. Bers. An extremal problem for quasiconformal maps and a theoremby Thurston. Acta Math. 141(1978), 73–98.

[Br] J. Brock. Iteration of mapping classes on a Bers slice: examples of alge-braic and geometric limits of hyperbolic 3-manifolds. In Lipa’s Legacy(New York, 1995), volume 211 of Contemp. Math., pages 81–106. Amer.Math. Soc., 1997.

[Bus] P. Buser. Geometry and Spectra of Compact Riemann Surfaces.Birkhauser Boston, 1992.

[CS] G. Cornell and J. H. Silverman, editors. Arithmetic Geometry. Springer-Verlag, 1986.

24

[DDT] H. Darmon, F. Diamond, and R. Taylor. Fermat’s last theorem. InCurrent Developments in Mathematics, 1995 (Cambridge, MA), pages1–154. Internat. Press, 1994.

[De1] P. Deligne. Preuve des conjectures de Tate et Shafarevitch [d’apres G.Faltings]. In Seminaire Bourbaki, 1983/84, pages 25–41. Asterisque,volume 121–122, 1985.

[De2] P. Deligne. Un theoreme de finitude pour la monodromie. In DiscreteGroups in Geometry and Analysis (New Haven, Conn., 1984), pages1–19. Birkhauser, 1987.

[Fal] G. Faltings. Endlichkeitssatze fur abelsche Varietaten uber Zahlkorpern.Invent. math. 73(1983), 349–366.

[FLP] A. Fathi, F. Laudenbach, and V. Poenaru. Travaux de Thurston sur lessurfaces. Asterisque, volume 66–67, 1979.

[Gd] F. Gardiner. Teichmuller Theory and Quadratic Differentials. WileyInterscience, 1987.

[Gr] H. Grauert. Mordells Vermutung uber rationale Punkte auf algebrais-chen Kurven und Funktionenkorper. IHES Publ. Math. 25(1965), 131–149.

[GH] P. Griffiths and J. Harris. Principles of Algebraic Geometry. WileyInterscience, 1978.

[IS] Y. Imayoshi and H. Shiga. A finiteness theorem for holomorphic familiesof Riemann surfaces. In Holomorphic Functions and Moduli II, pages207–219. Springer-Verlag: MSRI publications volume 11, 1988.

[IT] Y. Imayoshi and M. Taniguchi. An Introduction to Teichmuller Spaces.Springer-Verlag, 1992.

[Ka] N. Katz. An overview of Deligne’s proof of the Riemann hypothesis forvarieties over finite fields. In Mathematical Developments Arising fromHilbert Problems, volume 28 of Proc. Symp. Pure Math., pages 275–305.Amer. Math. Soc., 1976.

[Ko] K. Kodaira. A certain type of irregular algebraic surface. J. d’AnalyseMath. 19(1967), 207–215.

[La1] S. Lang. Integral points on curves. IHES Publ. Math. 6(1960), 27–43.

[La2] S. Lang. Introduction to Arakelov theory. Springer-Verlag, 1988.

[Le] O. Lehto. Univalent functions and Teichmuller spaces. Springer-Verlag,1987.

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[Lit] J. E. Littlewood. From Fermat’s Last Theorem to the Abolition ofCapital Punishment. In B. Bollabas, editor, Littlewood’s Miscellany.Cambridge University Press, 1986.

[Man] Y. Manin. A proof of the analog of the Mordell conjecture for algebraiccurves over function fields. Soviet Math. Dokl. 152(1963), 1061–1063.

[Maz1] B. Mazur. Notes on etale cohomology of number fields. Ann. Sci. Ec.Norm. Sup. 6(1973), 521 – 556.

[Maz2] B. Mazur. Arithmetic on curves. Bull. Amer. Math. Soc. 14(1986),207–259.

[Mc1] C. McMullen. Rational maps and Kleinian groups. In Proceedings of theInternational Congress of Mathematicians Kyoto 1990, pages 889–900.Springer-Verlag, 1991.

[Mc2] C. McMullen. Renormalization and 3-Manifolds which Fiber over theCircle, volume 142 of Annals of Math. Studies. Princeton UniversityPress, 1996.

[Mo] S. Mochizuki. The intrinsic Hodge theory of p-adic hyperbolic curves.In Proceedings of the International Congress of Mathematicians, Vol. II(Berlin, 1998), pages 187–196. Doc. Math., 1998.

[Mum] D. Mumford. A remark on Mahler’s compactness theorem. Proc. Amer.Math. Soc. 28(1971), 289–294.

[Nag] S. Nag. The Complex Analytic Theory of Teichmuller Space. Wiley,1988.

[NN] M. S. Narasimhan and M. V. Nori. Polarisations on an abelian variety.Proc. Indian Acad. Sci. Math. Sci. 90(1981), 125–128.

[Ot] J.-P. Otal. Le theoreme d’hyperbolisation pour les varietes fibrees dedimension trois. Asterisque, volume 235, 1996.

[Par] A. N. Parshin. Algebraic curves over function fields. Soviet Math. Dokl.183(1968), 524–526.

[Rez1] A. Reznikov. Three-manifolds class field theory. Selecta Math. 3(1997),361–399.

[Rez2] A. Reznikov. Hakenness and b1. Preprint, 1998.

[Ri] K. Ribet. Galois representations and modular forms. Bull. Amer. Math.Soc. 32(1995), 375–402.

[RS] K. Rubin and A. Silverberg. A report on Wiles’ Cambridge lectures.Bull. Amer. Math. Soc. 31(1994), 15–38.

26

[Sn] L. Schneps, editor. The Grothendieck theory of dessins d’enfants (Lu-miny, 1993), volume 200 of London Math. Soc. Lecture Note Ser. Cam-bridge Univ. Press, 1994.

[So] C. Soule. Lectures on Arakelov geometry. Cambridge University Press,1992.

[Sz] L. Szpiro. La conjecture de Mordell [d’apres G. Faltings]. In SeminaireBourbaki, 1983/84, pages 83–104. Asterisque, volume 121–122, 1985.

[TW] R. Taylor and A. Wiles. Ring-theoretic properties of certain Hecke al-gebras. Ann. of Math. (2) 141(1995), 553–572.

[Th1] W. P. Thurston. On the geometry and dynamics of diffeomorphisms ofsurfaces. Bull. Amer. Math. Soc. 19(1988), 417–432.

[Th2] W. P. Thurston. Hyperbolic structures on 3-manifolds II: Surface groupsand 3-manifolds which fiber over the circle. Preprint, 1986.

[Voj1] P. Vojta. Diophantine Approximations and Value Distribution Theory,volume 1239 of Lecture Notes in Mathematics. Springer-Verlag, 1987.

[Voj2] P. Vojta. Siegel’s theorem in the compact case. Ann. of Math.133(1991), 509–548.

[Wl1] A. Weil. On discrete subgroups of Lie groups. Annals of Math. 72(1960),369–384.

[Wl2] A. Weil. Sur l’analogie entre les corps de nombres algebriques et lescorps de fonctions algebriques [1939a]. In Oeuvres Scient., volume I,pages 236–240. Springer-Verlag, 1980.

[Wi] A. Wiles. Modular elliptic curves and Fermat’s last theorem. Ann. ofMath. 141(1995), 443–551.

[Wol] S. Wolpert. Geodesic length functions and the Nielsen problem. J. Diff.Geom. 25(1987), 275–296.

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